Evolution of hybrid quantum-classical wavefunctions

DOI10.1016/J.PHYSD.2022.133450arXiv2112.12144WikidataQ114141880 ScholiaQ114141880MaRDI QIDQ6386470

Publication date: 22 December 2021

Abstract: A gauge-invariant wave equation for the dynamics of hybrid quantum-classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge invariance in the variational principle underlying a hybrid wave equation previously proposed by the authors. While the original construction ensures a positive-definite quantum density matrix, the present model also guarantees the same property for the classical Liouville density. After a suitable wavefunction factorization, gauge invariance is achieved by resorting to the classical Lagrangian paths made available by the Madelung transform of Koopman wavefunctions. Due to the appearance of a phase-space analogue of the Berry connection, the new hybrid wave equation is highly nonlinear and it is proposed here as a platform for further developments in quantum-classical dynamics. Indeed, the associated model is Hamiltonian and appears to be the first to ensure a series of consistency properties beyond positivity of quantum and classical densities. For example, the model possesses a quantum-classical Poincar'e integral invariant and its special cases include both the mean-field model and the Ehrenfest model from chemical physics.

Mathematics Subject Classification ID

PDEs in connection with biology, chemistry and other natural sciences (35Q92) Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators (47A68) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20) Covariant wave equations in quantum theory, relativistic quantum mechanics (81R20) Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory (81Q70) Quantum state spaces, operational and probabilistic concepts (81P16)

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